Event Horizon and Maximal Extension

Derivation

Event Horizon and Maximal Extension

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Statement

Starting from the Schwarzschild vacuum solution, we show that the locus \(r = 2GM\) (units \(c=1\)) is a coordinate singularity, not a curvature singularity: the Kretschmann invariant is finite there while the Schwarzschild chart degenerates. Introducing the tortoise coordinate and Eddington–Finkelstein null coordinates, and then the exponential Kruskal–Szekeres coordinates \((U,V)\) with \(\kappa = 1/4GM\), we obtain a metric \(ds^2 = -\tfrac{32\,G^3M^3}{r}\,e^{-r/2GM}\,dU\,dV + r^2\,d\Omega^2\) that is smooth and non-degenerate across \(r=2GM\). The single Schwarzschild exterior thereby extends to four regions — two asymptotically flat exteriors, a black-hole interior, and a white-hole interior — with the true curvature singularity confined to the hyperbola \(UV = 1\) (\(r=0\)).

Why it matters

The apparent blow-up of the Schwarzschild metric at \(r=2GM\) misled a generation into thinking a physical wall or singularity sat at the "Schwarzschild radius." Constructing Kruskal–Szekeres coordinates is the cleanest proof that the horizon is a perfectly regular null surface an infalling observer crosses in finite proper time — the pathology lived in the chart, not the spacetime.

The maximally extended diagram also fixes the global causal structure: it exhibits the horizon as a one-way membrane, reveals the white-hole and second-exterior regions forced by the field equations, and provides the geometric backbone for surface gravity, Hawking temperature, and the Penrose–Carter conformal picture.

Assumptions

Vacuum, static, spherically symmetric spacetime (Schwarzschild).Drop it and Birkhoff's theorem no longer pins the metric; rotation (Kerr) or charge (Reissner–Nordström) changes the horizon structure and the extension has different global topology. Geometric units \(c=1\); \(M\) is the ADM mass and \(M>0\).With \(M\le 0\) there is no horizon (naked singularity at \(r=0\)); restoring \(c\) simply rescales \(r_s = 2GM/c^2\) but changes none of the argument. The exterior chart \(r>2GM\) is a genuine coordinate patch, and curvature invariants are chart-independent scalars.If one forgets that scalars like \(R_{abcd}R^{abcd}\) are invariants, one cannot distinguish a coordinate degeneracy from a real singularity, and the whole regularity argument collapses. Analytic (maximal) extension is the physically privileged one.Drop maximality and the manifold is geodesically incomplete: timelike/null geodesics run off the edge of the patch at finite affine parameter with curvature still finite — an unphysical "hole" in spacetime.

Derivation

\[ ds^2 = -\left(1-\frac{2GM}{r}\right)dt^2 + \left(1-\frac{2GM}{r}\right)^{-1}dr^2 + r^2\,d\Omega^2,\qquad d\Omega^2 = d\theta^2 + \sin^2\theta\,d\varphi^2 \]

Schwarzschild line element, assumed from the vacuum field equations. Write \(f(r)\equiv 1-2GM/r\). A

\[ g_{tt}=-f(r)\to 0,\qquad g_{rr}=f(r)^{-1}\to\infty \quad\text{as } r\to 2GM^{+} \]

The metric components degenerate at \(r=2GM\): \(g_{tt}\) vanishes and \(g_{rr}\) diverges, so \(\det g \to\) finite but the chart is singular. This alone does not decide whether the geometry is singular. A

\[ K \equiv R_{abcd}R^{abcd} = \frac{48\,G^2M^2}{r^6} \;\xrightarrow{\,r=2GM\,}\; \frac{48\,G^2M^2}{(2GM)^6} = \frac{3}{4\,G^4M^4} \;<\;\infty \]

The Kretschmann scalar is a coordinate-invariant. It is finite at \(r=2GM\) and diverges only as \(r\to 0\). Hence \(r=2GM\) is a coordinate singularity and \(r=0\) is the genuine curvature singularity. C

\[ dr_* \equiv \frac{dr}{f(r)} = \frac{dr}{1-\tfrac{2GM}{r}} = \frac{r\,dr}{r-2GM} \]

Define the tortoise coordinate to absorb the \(g_{rr}\) divergence. Radial null geodesics will satisfy \(dt = \pm dr_*\). B

\[ r_* = r + 2GM\ln\!\left|\frac{r}{2GM}-1\right| + \text{const} \]

Integrate step 4 by writing \(\dfrac{r}{r-2GM}=1+\dfrac{2GM}{r-2GM}\). Note \(r_*\to-\infty\) as \(r\to2GM^{+}\): the horizon is pushed to infinite tortoise distance. B

\[ u \equiv t - r_*,\qquad v \equiv t + r_*,\qquad ds^2 = -f(r)\,du\,dv + r^2\,d\Omega^2 \]

Pass to advanced/retarded null coordinates. Radial light rays are \(u=\text{const}\) or \(v=\text{const}\); the metric is off-diagonal but \(f(r)\) still vanishes at the horizon, so \((u,v)\) does not yet cover it. B

\[ U \equiv -e^{-\kappa u},\qquad V \equiv e^{\kappa v},\qquad \kappa \equiv \frac{1}{4GM} \quad(\text{region I: } U<0,\;V>0) \]

Exponentiate the null coordinates. The constant \(\kappa\) (the surface gravity) is fixed by demanding the horizon map to a finite value: \(r\to2GM\) means \(u\to+\infty,\,v\to-\infty\), giving \(U\to0^-,\,V\to0^+\). C

\[ dU\,dV = \kappa^2 e^{\kappa(v-u)}\,du\,dv = \kappa^2 e^{r_*/2GM}\,du\,dv \;\Rightarrow\; ds^2 = -\frac{32\,G^3M^3}{r}\,e^{-r/2GM}\,dU\,dV + r^2\,d\Omega^2 \]

Substitute \(du\,dv=\kappa^{-2}e^{-r_*/2GM}dU\,dV\) with \(e^{r_*/2GM}=\left(\tfrac{r}{2GM}-1\right)e^{r/2GM}\) and \(f=\tfrac{r-2GM}{r}\); the diverging \(f\) cancels the vanishing factor. The prefactor \(\tfrac{32G^3M^3}{r}e^{-r/2GM}\) is smooth and strictly positive at \(r=2GM\). The metric is regular across the horizon. C

\[ UV = -\left(\frac{r}{2GM}-1\right)e^{r/2GM},\qquad \frac{V}{U} = -\,e^{t/2GM} \]

Multiply and divide the definitions of \(U,V\). The first relation defines \(r=r(UV)\) implicitly and smoothly for \(UV<1\); the second recovers the Schwarzschild time \(t\) as a boost parameter in the \((U,V)\) plane. B

\[ UV=0 \;\Leftrightarrow\; r=2GM \ (\text{horizon}),\qquad UV=1 \;\Leftrightarrow\; r=0 \ (\text{singularity}) \]

Since \(r(UV)\) is well defined for all \(UV<1\), the coordinates \((U,V)\in\mathbb{R}^2\) (with \(UV<1\)) analytically extend region I to four regions: exteriors I \((U<0,V>0)\) and III \((U>0,V<0)\), black-hole interior II \((U>0,V>0)\), white-hole interior IV \((U<0,V<0)\). \(r=0\) is the spacelike hyperbola \(UV=1\), a future branch \((V>0)\) and a past branch \((V<0)\). C

Result

\[ ds^2 = -\frac{32\,G^3M^3}{r}\,e^{-r/2GM}\,dU\,dV + r^2(U,V)\,d\Omega^2,\qquad UV = \left(1-\frac{r}{2GM}\right)e^{r/2GM} \]

Reading. In Kruskal–Szekeres coordinates the metric coefficient is finite and non-zero at \(r=2GM\), so the horizon is a smooth null surface, not an edge of spacetime. Radial light cones stay at \(45^\circ\) everywhere (lines \(U=\text{const}\), \(V=\text{const}\)), making the causal structure manifest: inside region II every future-directed causal curve has increasing \(r\)-decreasing character and must hit \(UV=1\) (\(r=0\)). The horizon \(UV=0\) is crossed by an infaller in finite proper time, while a static exterior observer sees the crossing red-shifted to \(t\to\infty\).

Units check. With \(c=1\), \([GM]=\text{length}\), so \(r\), \(GM\) share dimensions and \(U,V\) are dimensionless (arguments of \(e^{\kappa u}\), \(\kappa=1/4GM\), \([\kappa u]=1\)). The prefactor \(G^3M^3/r\) has dimension \(\text{length}^3/\text{length}=\text{length}^2\), so \((\text{length}^2)\,dU\,dV\) is a length-squared: a valid \(ds^2\). Restoring \(c\): \(r_s = 2GM/c^2\), \(\kappa = c^4/4GM\) (acceleration).

Limiting cases

Far field \(r\gg 2GM\): \(UV\to -\tfrac{r}{2GM}e^{r/2GM}\to-\infty\), light cones open into ordinary Minkowski null lines; the metric reduces to flat spacetime and region I is asymptotically flat.
Weak field / Newtonian: \(f\approx 1-2GM/r\) gives \(g_{tt}\approx-(1+2\Phi)\) with \(\Phi=-GM/r\); the horizon disappears to \(r\to0\) as \(GM\to0\), recovering special relativity.
On the horizon \(r=2GM\): \(UV=0\); surface gravity \(\kappa=1/4GM\) is constant over the surface (zeroth law), and \(T_H=\kappa/2\pi=1/8\pi GM\) in units \(\hbar=k_B=1\).
Near-horizon Rindler limit: writing \(\rho^2\propto(r-2GM)\), the \((t,r)\) part becomes \(ds^2\approx -\kappa^2\rho^2\,dt^2+d\rho^2\), flat spacetime in Rindler coordinates — the horizon is locally an acceleration horizon.
Extremal-mass limit \(M\to\infty\): horizon curvature \(K(2GM)=3/4G^4M^4\to0\); a supermassive horizon is locally almost flat and tidally gentle to cross.

Breaks when

Rotation or charge. For Kerr (\(a\neq0\)) or Reissner–Nordström (\(Q\neq0\)) the transformation above fails: there are two horizons \(r_\pm\), each needs its own Kruskal-type patch, the maximal extension has infinitely many regions and an inner Cauchy horizon (unstable to mass inflation). The single-\(\kappa\) exponential map does not globally cover them.
Dynamical / non-vacuum spacetimes. If \(M=M(v)\) (accretion, evaporation) or \(T_{ab}\neq0\), the metric is not static, \(\partial_t\) is not Killing, \(\kappa\) is not constant, and the tortoise integral in step 5 is ill-defined; one must use apparent/trapping horizons instead of the Killing horizon \(UV=0\).
At the true singularity \(r=0\) (\(UV=1\)). The Kretschmann scalar diverges, \(r(UV)\) ceases to be smooth, and no coordinate change removes it — the extension terminates on the spacelike singularity, which is the genuine boundary.
Cosmological horizon present (\(\Lambda\neq0\)). Schwarzschild–de Sitter adds a second Killing horizon at large \(r\); the asymptotic region is no longer flat and the global Kruskal diagram must be tiled differently.

Failure modes

Calling \(r=2GM\) a singularity. Reading the divergent \(g_{rr}\) as physical, without checking the invariant \(K\); the correct diagnostic is a coordinate-independent scalar, not a metric component.
Dropping the sign / branch of \(U\). Writing \(U=+e^{-\kappa u}\) in region I; the exterior requires \(U<0\), and forgetting this collapses regions I and III or mislabels the interior.
Using the wrong \(\kappa\). Choosing any constant other than \(\kappa=1/4GM\) leaves a residual conformal factor that still vanishes or diverges at the horizon, so the metric is not regularized. The value is forced by the surface gravity.
Treating \(t\) as time inside the horizon. For \(r<2GM\), \(f<0\): \(r\) becomes timelike and \(t\) spacelike. Students who keep \(t\) as "time" wrongly conclude one can hover at fixed \(r\).
Confusing coordinate crossing time with proper time. Concluding the infaller "never reaches" the horizon because Schwarzschild \(t\to\infty\); proper time to cross is finite, as the Kruskal chart shows directly.
Forgetting the constant of integration in \(r_*\). It shifts \(U,V\) by a multiplicative constant and is harmless for the geometry, but mixing conventions between sources corrupts numerical values of \(UV\).

Discussion

The construction reframes what "singularity" means. A metric component is a chart-dependent object; only scalars built from the Riemann tensor (\(R_{abcd}R^{abcd}\), \(R_{ab}R^{ab}\), etc.) carry invariant content. Because \(K=48G^2M^2/r^6\) is finite at \(r=2GM\), no observer measures divergent tidal forces there — for a solar-mass hole the horizon curvature is a modest \(\sim10^{-13}\,\text{m}^{-4}\). The Schwarzschild "singularity" was a failure of the coordinate patch to be geodesically complete, cured by analytic extension.

The Kruskal diagram makes the horizon's one-way character geometric. Since radial null rays are \(45^\circ\) lines, the future light cone of any event in region II points entirely toward larger \(V\) and \(U\), i.e. toward \(UV=1\); escape to region I would require moving outside the light cone. The horizon \(U=0\) (future) is a null surface generated by the outgoing rays that asymptote to it — a Killing horizon of \(\partial_t\), whose norm \(\xi^a\xi_a=-f\) vanishes there, and whose surface gravity \(\kappa=\tfrac12 f'(2GM)=1/4GM\) sets the Hawking temperature.

The full maximal extension contains structure with no counterpart in the collapse of a real star: region III is a second asymptotically flat universe causally disconnected from ours, and region IV is a white hole from which signals can only emerge. The two exteriors are joined by an Einstein–Rosen bridge — a non-traversable wormhole whose throat pinches off faster than any timelike curve can cross, as one proves by studying the \(t=\text{const}\) spatial slices \(V=-U\) of the geometry. For an astrophysical black hole formed by collapse, regions III and IV are replaced by the collapsing matter's interior, so only regions I and II are physical; the eternal solution is nonetheless the correct vacuum exterior and the arena for Hawking's derivation.

Common misconceptions. "Nothing can reach the horizon" confuses coordinate time with proper time. "The horizon is where escape velocity equals \(c\)" is a Newtonian heuristic that gives the right \(r_s\) for the wrong reason — the horizon is a global causal boundary, not a local speed condition. And the Einstein–Rosen bridge is not a shortcut: it is spacelike at its throat and non-traversable classically.

Worked examples

Schwarzschild radius of the Sun. \[ r_s = \frac{2GM_\odot}{c^2} \]

Restore SI units; the horizon sits at the coordinate singularity \(r=r_s\). Symbols first, then numbers. A

\[ r_s = \frac{2(6.674\times10^{-11}\,\text{m}^3\text{kg}^{-1}\text{s}^{-2})(1.989\times10^{30}\,\text{kg})}{(2.998\times10^{8}\,\text{m s}^{-1})^2} \]

Insert \(G\), \(M_\odot\), \(c\). Numerator \(=2.655\times10^{20}\,\text{m}^3\text{s}^{-2}\); denominator \(=8.988\times10^{16}\,\text{m}^2\text{s}^{-2}\). A

\[ r_s \approx 2.95\times10^{3}\,\text{m} \approx 2.95\ \text{km} \]

Reading. If the Sun were compressed inside \(\sim3\,\text{km}\), its surface would lie at the horizon. The horizon is a coordinate singularity, so this radius marks a causal boundary, not a material surface. Units: \((\text{m}^3\text{kg}^{-1}\text{s}^{-2}\cdot\text{kg})/(\text{m}^2\text{s}^{-2})=\text{m}\). Check.

Tidal curvature at the horizon of a solar-mass hole (regularity check). \[ K(r) = \frac{48\,G^2M^2}{c^4\,r^6},\qquad K(r_s)=\frac{48\,G^2M^2}{c^4 (2GM/c^2)^6}=\frac{3\,c^8}{4\,G^4M^4} \]

Evaluate the invariant Kretschmann scalar exactly at \(r=r_s\); a finite answer proves the horizon is regular. Symbols before numbers. C

\[ K(r_s) = \frac{3\,(2.998\times10^{8})^{8}}{4\,(6.674\times10^{-11})^{4}(1.989\times10^{30})^{4}} \]

Insert numbers: \(c^8=6.53\times10^{67}\,\text{m}^8\text{s}^{-8}\), \(G^4=1.98\times10^{-41}\), \(M^4=1.57\times10^{121}\,\text{kg}^4\). B

\[ K(r_s) \approx 1.6\times10^{-13}\ \text{m}^{-4}\quad(\text{finite}),\qquad K(r\to0)\to\infty \]

Reading. The curvature at the solar-mass horizon is minuscule and finite — an infalling astronaut feels negligible tidal stress crossing it — confirming \(r=2GM\) is a coordinate singularity. Only at \(r=0\) does \(K\) diverge, the genuine singularity at \(UV=1\). Units: \(c^8/(G^4M^4)\) has dimension \((\text{m}^8\text{s}^{-8})/(\text{m}^{12}\text{kg}^{-4}\text{s}^{-8}\cdot\text{kg}^4)=\text{m}^{-4}\). Check.

Problems

Surface gravity and Hawking temperature. Show \(\kappa=\tfrac12 f'(r_s)\) for \(f=1-2GM/r\) and compute \(T_H=\hbar\kappa/2\pi c\,k_B\) (SI) for a solar-mass black hole.
Solution
\(f'(r)=2GM/r^2\), so \(\kappa=\tfrac12 c^2 f'(r_s)=\tfrac12 c^2\cdot\tfrac{2GM}{(2GM/c^2)^2}=\tfrac{c^4}{4GM}\). Numerically \(\kappa=\tfrac{(2.998\times10^8)^4}{4(6.674\times10^{-11})(1.989\times10^{30})}=\tfrac{8.08\times10^{33}}{5.31\times10^{20}}\approx1.52\times10^{13}\,\text{m s}^{-2}\). Then \(T_H=\tfrac{\hbar\kappa}{2\pi c k_B}=\tfrac{(1.055\times10^{-34})(1.52\times10^{13})}{2\pi(2.998\times10^8)(1.381\times10^{-23})}\approx6.2\times10^{-8}\,\text{K}\).
Tortoise value. Evaluate \(r_*\) at \(r=3GM\) (units \(c=1\)) taking the integration constant zero, and state the sign of \(UV\) there.
Solution
\(r_*=r+2GM\ln|r/2GM-1|=3GM+2GM\ln(3/2-1)=3GM+2GM\ln(0.5)=3GM-2GM(0.693)=3GM-1.386GM=1.614GM\). Since \(r=3GM>2GM\) this is region I, \(f>0\), and \(UV=(1-r/2GM)e^{r/2GM}=(1-1.5)e^{1.5}=-0.5(4.482)=-2.24<0\). Negative, consistent with an exterior point.
Proper radial infall is finite. For a radial geodesic dropped from rest at infinity, \(dr/d\tau=-\sqrt{2GM/r}\) (units \(c=1\)). Find the proper time to fall from \(r=2GM\) to \(r=0\).
Solution
\(\tau=\int_0^{2GM}\tfrac{dr}{\sqrt{2GM/r}}=\tfrac{1}{\sqrt{2GM}}\int_0^{2GM} r^{1/2}dr=\tfrac{1}{\sqrt{2GM}}\cdot\tfrac{2}{3}(2GM)^{3/2}=\tfrac{2}{3}(2GM)=\tfrac{4GM}{3}\). Finite — the interior is traversed in finite proper time even though Schwarzschild \(t\) diverges at the horizon. Restoring units, \(\tau=4GM/3c^3\).
Locate \(r=0\) in Kruskal coordinates. Using \(UV=(1-r/2GM)e^{r/2GM}\), find the curve of the singularity and classify it.
Solution
Set \(r=0\): \(UV=(1-0)e^{0}=1\). The singularity is the hyperbola \(UV=1\), i.e. \(V=1/U\), with two branches: future \((U>0,V>0)\) and past \((U<0,V<0)\). Its tangent lies inside the light cone (spacelike surface), so \(r=0\) is a spacelike singularity — a moment of time inside the hole, not a place, which is why it cannot be avoided once in region II.
Regularity of the prefactor. Show the Kruskal metric coefficient \(P(r)=32G^3M^3 e^{-r/2GM}/r\) is finite and nonzero at \(r=2GM\), and give its value for \(M=M_\odot\) (SI, note it carries a \(c\)-dependent dimensionful factor; work in \(c=1\)).
Solution
At \(r=2GM\): \(P=32G^3M^3 e^{-1}/(2GM)=16G^2M^2 e^{-1}\approx5.89\,G^2M^2\), manifestly finite and positive. In geometric units \(GM_\odot=1.48\times10^3\,\text{m}\), so \(P=16(1.48\times10^3)^2 e^{-1}=16(2.19\times10^6)(0.368)\approx1.29\times10^7\,\text{m}^2\). Because \(P\neq0\) and \(\det\) of the 2D block \(=-P^2/4\neq0\), the metric is non-degenerate across the horizon, proving \(r=2GM\) is regular.